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Understanding Symmetries and Geometry through Dance

Understanding Symmetries and Geometry through Dance. Laura Olliverrie. Motivation.

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Understanding Symmetries and Geometry through Dance

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  1. Understanding Symmetries and Geometry through Dance Laura Olliverrie

  2. Motivation • The link between math and music has already been acknowledged, but lets take a step further to understand the connection between math and dance. I will explore the connections between mathematics and dance in both practice and theory. • Dancing Mathematics and the Mathematics of Dance by Sarah-Marie Belcastro and Karl Schaffer

  3. Introduction • Every dance choreography incorporates the use of angles, geometric shapes, parallels, patterns, and symmetry in some way. • Dancers use symmetry and geometry to improve their performances and make them visually appealing. • Let us think of dance as an artistic endeavor and look at how deep mathematical concepts can be viewed through dance.

  4. Superficial Links • Counting Steps • Music is divided into counts and dancers use counting to mark the times at which movements are done • Noticing Shapes

  5. Symmetry • The human body lacks rotational symmetry so it is impossible for a dancer to continue spinning forever. They will lose their balance. • Humans do have reflection symmetry • In dance, we notice the use of symmetry of motion for an individual dancer and that of a group

  6. Klein Four Group • Dance choreography often displays pattern symmetry types such as reflection, rotation, translation and glide • Using just four symmetries—translation, mirror reflection, 180-degree rotation, and glide reflection—we can create what is called the Klein four group by combining the symmetries pairwise.

  7. Try it Out! • If we are to fill in the table with compositions of translation (T), glide reflection (G), 180-degree rotation (R), and mirror reflection (M), we will end up with the multiplication table for the Klein four group.

  8. Terminology of Symmetries in Time and Space • Unison (a set of movements performed at the same time) • Canon (a set of movements offset in time from each other) • Inversion (movements performed in reverse sequence) • Retrograde (each movement is reversed, in addition to the sequence being reversed)

  9. How Math Sounds? • Rhythm is an important part of any dance, and some dance forms such as tap and clogging use sound extensively as well. • Let one person claps the eight-beat rhythm and the other clap the three-beat rhythm, each clapping loudly on the first beat of each cycle.

  10. https://www.youtube.com/watch?v=Ws2y-cGoWqQ

  11. Basic Physics • Velocity is how fast and in what direction an object is moving • Momentum is mass times velocity • Force is basically a push or a pull • Gravity, support from the floor and friction from the floor are the forces that act on a dancer

  12. Balance • No total force and no total torque (measure of how much a force acting on an object causes that object to rotate) • The center of gravity must remain directly above the area of contact with the floor

  13. Turns/Spins • The relevant quantities that describe turns/spins are: • Angular velocity is how fast an object spins • Rotational inertia is the inertia of a rotating object • Angular momentum is rotational inertia times angular velocity

  14. Comparison • Motion without spins • Velocity • Mass • Momentum • Force • Motion with spins • Angular velocity • Rotational inertia • Angular momentum • Torque

  15. Other Dance Forms Western Ballet and BharatyaNatyam - Use a strong sense of line Flamenco Duet - 180-degree rotational symmetry Hip Hop Dance - Angles and geometric shapes

  16. Pedagogical Applications to Math in Dance • A number of programs have arised that are encouraging mathematical thought through movement. • Teaching math through dance • Bridge the gap • Helps kids combat math anxiety • Maths Dance: Triangular Squares by Corinne Wolfe https://www.youtube.com/watch?v=Om4j8Nt2JWQ

  17. Further Links • In an effort to attract more girls to a STEM field of study, Virtual Environment Interactions (VEnvl) software allows girls top program 3-D characters to perform dance moves just by moving their own bodies. • The girls are then encourages to develop new computing strategies to improve their choreography.

  18. References • https://www.jstor.org/stable/pdf/10.4169/194762111x12954578042939.pdf?refreqid=excelsior%3Adb17025fbc4fa70d7cedac450cc2e68c • Schaffer, K., Stern, E. and Kim, S. “Math Dance” MoveSpeakSpin, Santa Cruz 2001.

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